Quadratic Approximation for Singular Integrals

(1)

Journal of Mathematics and Statistics 8 (2): 292-295, 2012 ISSN 1549-3644

292

Quadratic Approximation for Singular Integrals

Mostefa Nadir

Department of Mathematics, Faculty of Mathematics and Informatics

Laboratory of Pure and Applied Mathematics, University of Msila 28000, Algeria

Abstract: In this study we present a new approach based on a quadratic approximation for singular integrals of Cauchy type, by using a small technical we arrive to eliminate completely the singularity of this integral. Noting that, this approximation is destined to solve numerically all singular integral equations with Cauchy kernel type on an oriented smooth contour.

Key words: Quadratic approximation, singular integral equations, oriented smooth contour, eliminate completely, quadratic method, singular integral operator

INTRODUCTION

Many problems of mathematical physics, engineering and contact problems in the theory of elasticity lead to singular integral equations with Cauchy kernel type: Eq. 1:

0

0 0 0 0

0

b(t ) (t)

a(t ) (t ) dt k(t, t ) (t)dt f (t )

i Γt t Γ

ϕ

ϕ + + ϕ =

π

∫

−

∫

(1)

where, Γ designates an oriented smooth contour, the points t and t0 are on Γ.

The Eq. 1 plays an important role in modern numerical computations in the applied sciences, in particular in the applied mathematical.

Our schemes describe the quadratic method for the approximation of singular integral operator with Cauchy kernel Eq. 2:

0 0

0

1 (t)

F(t ) dt, t, t

i Γt t

ϕ

= ∈Γ

π

∫

− (2)

by a sequence of numerical integration operators.

Noting that, for the existence of the principal value of the integral (2) for a given density ϕ(t) we will need more than mere continuity. In other words, the density

ϕ(t) has to satisfy the Hölder condition H(µ) (Muskhelishvili, 2008).

The function ϕ(t) will be said to satisfy a Hölder condition on Γ if for any two points t1 and t2 of Γ:

2 1 2 1

| (t )ϕ − ϕ(t ) | A | t≤ −t | , 0µ < µ ≤1

where, A is positive constant, called the Holder constant and µ the Holder index.

The quadrature: Noting by t the parametric complex function t(s) of the curve Γ defined by:

t(s)=x(s)+iy(s), a≤ ≤s b

where, x(s) and y(s) are continuous functions on the finite interval of definition [a, b] and have a continuous first derivatives x’(s) and y’(s) never simultaneously null. Let N be an arbitrary natural number, generally we take it large enough and divide the interval [a, b] into N equal subintervals of [a, b]:

0 1 N

[a, b] {a= = < <s s ...<s =b}

be called I1 to IN so that, we have Iσ+1 [sσ, sσ+1]:

l

s a , l b a , 0,1, 2,...., N

N

σ= + σ = − σ =

Further, fixing a natural number M>1 and divide each of segments [sσ, sσ+1] by the equidistant points:

k

h l

s s k , h , k 0,1,..., 2M

2M N

σ = +σ = =

In other words, we have for each subinterval [sσ,sσ+1] the following subdivision:

1 0 1 2M 1

[s ,s_σ _σ+] {s= _σ=s_σ <s_σ <...<s_σ =s_σ+}

Denoting by:

k k

t_σ=t(s ), t_σ _σ =t(s ); _σ σ =0,1, 2,..., N; k=0,1,...., 2M

Assuming that for the indices σ,v = 0,1,2,…N-1 the points t and t0 belong respectively to the arcs t tσ σ+1

(2)

J. Math. & Stat., 8 (2): 292-295, 2012

293 and t tν ν+1

⌢

where t tα α+1 ⌢

designates the smallest arc with ends ta and ta+1 (Nadir, 1985; 1998; Nadir and Antidze,

2004; Nadir and Lakehali, 2007; Sanikidze, 1970; Antidze, 1975).

For an arbitrary number σ = 0,1,2,…N-1 we define the spline function S2(ϕ,t,σ) dependents of ϕ, t and σ

which represents the quadratic approximation of the function density ϕ(t) on the subinterval [tσ, tσ+1] of the

curve Γ. As we know, the interval [tσ, tσ+1] is divided

into a subintervals [tσk, tσ(k+2)] of length (tσ(k+2)-tσk) k =

2i, I = 0,1,...,M-1. We interpolate the function density ϕ(t) with respect to the values ϕ(tσk), ϕ(tσ(k+1)) and ϕ(tσ(k+2)) at

the points tσk, tσ(k+1) and tσ(k+2) respectively with a quadratic

polynomial, given by the following formula. For tok≤t≤σ(k+2) Eq. 3:

(k 1) ( k 2)

2 k

(k 1) k ( k 2) k k (k 2)

(k 1) ( k 1) k (k 2) (k 1)

k ( k 1)

(k 2) (k 2) k (k 2) ( k 1)

(t t )(t t )

S ( ; t, ) (t )

(t t )(t t )

(t )

(t t )(t t )

(t )

(t t )(t t )

σ + σ + σ σ + σ σ + σ

σ σ +

σ + σ + σ σ + σ +

σ σ +

σ + σ + σ σ + σ +

− −

ϕ σ = ϕ

− − − − − ϕ − − − − + ϕ − − (3)

This spline function exists and is unique also called a quadratic interpolating polynomial.

Define for an arbitrary numbers σ and v such that 0≤σ,v≤N-1 the function βσv (ϕ,t,t0) dependents of ϕ,t

and t0 by Eq. 4:

0 2 2 0

( , t, t ) S ( , t, ) S ( , t , )

σν

β ϕ = ϕ σ − ϕ ν (4)

where, t∈[tσ, tσ+1] and t0 ∈ [tv,tv+1] and noting by Ψσv

(ϕ,t,t0) the quadratic approximation of the density ϕ(t)

at the point t∈[tσ,tσ+1] Eq. 5:

0 0 0

( , t, t ) (t ) ( , t, t )

σν σν

ψ ϕ = ϕ + β ϕ (5)

Replacing the density ϕ(t) by this last quadratic approximation Ψσv (ϕ, t, t0) in the singular integral (2):

0

1 ( t ) F ( t ) d t

i Γt t

ϕ =

π

∫

−

In order to obtain the following approximation Eq. 6: 0 0 0 0 0 0

1 ( , t, t ) S( , t )

i t t 1 ( , t, t )

dt (t ) dt

i t t

σν Γ σν Γ ψ ϕ ϕ = π − β ϕ

= ϕ +

π −

∫

(6)

Theorem: Let Γ be an oriented smooth contour and let ϕ be a density function defined on Γ and satisfies the Hölder condition H(µ) then, the following estimation:

0 0

| F ( t ) S ( , t ) | m a x C ln ( 2 M N ) C

( , ); N , M 1 ( 2 M N )µ Nµ

− ϕ ≤

>

Holds for all t and t0 on the contour Γ. The constant

C depends only of Γ.

Proof: Choosing the points t∈[tσ,tσ+1] and t0∈[tv,tv+1]

then for tσk≤t≤tσ(k+2) and tik≤ t0≤ tv(k+2) we have Eq. 7:

0 0

(k 1) ( k 2) k ( k 1) k ( k 2) k

k ( k 2)

( k 1) ( k 1) k ( k 2) ( k 1)

k ( k 1)

( k 2) (k 2) k ( k 2) ( k 1)

( , t, t ) (t) (t )

(t t )(t t )

{ (t )

(t t )(t t )

(t )

(t t )(t t )

(t )

(t t )(t t )

(

σν

σ + σ + σ σ + σ σ + σ

σ σ +

σ + σ + σ σ + σ +

σ σ +

σ + σ + σ σ + σ + ϕ − ψ ϕ = ϕ − ϕ

− − − ϕ − − − − − ϕ − − − − + ϕ − −

− 0 (k 1) 0 (k 2) k ( k 1) k ( k 2) k

0 k 0 ( k 2)

( k 1) ( k 1) k (k 2) ( k 1)

0 k 0 (k 1)

( k 2) ( k 2) k ( k 2) (k 1)

t t )(t t )

(t )

(t t )(t t )

(t )

(t t )(t t )

(t )}

(t t )(t t )

ν + ν +

ν ν + ν ν + ν

ν ν +

ν + ν + ν ν + ν +

ν ν +

ν + ν + ν ν + ν +

− − _ϕ

− − − − + ϕ − − − − − ϕ − − (7)

Taking into account the expression (7) we get Eq. 8:

1 0 0 N 1 0 t t 0 0

1 (t) ( , t, t )

dt

i t t

1 (t) ( , t, t )

dt

i σ σ+ t t

σν Γ

−

σν σ=

ϕ − ψ ϕ ₌

π −

ϕ − ψ ϕ

π −

∫

∑∫

(8)

Hence:

2 k ( 2 k 2 )

N 1 M 1 ₀

0 0 0 k 0 t t

0 (k 1) (k 2)

k ( k 1) k (k 2) k

k ( k 2)

(k 1) (k 1) k ( k 2) (k 1)

k (k 1) (k 2)

1 (t) (t )

F(t ) S( , t )

i t t

(t t )(t t )

{ (t )

(t t )(t t ) (t t )(t t )

(t ) (t t )(t t )

(t t )(t t ) (t

σ σ +

− − σ= =

σ + σ + σ σ + σ σ + σ

σ σ +

σ + σ + σ σ + σ +

σ σ + σ +

ϕ − ϕ

− ϕ =

π − − − − ϕ − − − − − ϕ − − − − +

∑ ∑ ∫

( k 2) k (k 2) ( k 1) 0 ( k 1) 0 (k 2)

k ( k 1) k ( k 2) k

0 k 0 (k 2)

(k 1) (k 1) k ( k 2) ( k 1)

0 k 0 (k 1) ( k 2) k ( k 2) ( k 1)

(t ) t )(t t ) (t t )(t t )

(t ) (t t )(t t )

(t t )(t t )

(t ) (t t )(t t )

(t t )(t t ) (t t )(t t

σ + σ σ + σ + ν + ν +

ν ν + ν ν + ν

ν ν +

ν + ν + ν ν + ν +

ν ν +

ν + ν ν + ν +

ϕ − − − − − ϕ − − − − + ϕ − − − − −

− − (k 2)

0

1 (t )} dt )ϕ ν + t−t

It can be seen that, the equality t-t0 = 0 is possible

(3)

J. Math. & Stat., 8 (2): 292-295, 2012

294 integral (8) exists when t tends to t0. The last case, if σ

= v we can easily seeing that, the function βσσ (ϕ, t,t0)

contains (t-t0) as factor this means, for the points t,t0∈

[tσ,tσ+1] we write Eq. 9:

0 2 2 0

( , t, t ) S ( , t, ) S ( , t , )

σσ

β ϕ = ϕ σ − ϕ σ

Hence, for tσk≤t,t0≤ tσ(k+2):

0 (k 1) 0 (k 2)

0 k

(k 1) k (k 2) k

0 (k 2) 0 k

(k 1) (k 1) k (k 2) (k 1)

0 k 0 (k 1)

(k 2) k (k 2)

(t t )[(t t ) (t t )]

( ,t,t ) { (t )

(t t )(t t )

(t t )[(t t ) (t t )] (t )

(t t )(t t )

(t t )[(t t ) (t t )]

(t t )(t t

σ + σ +

σσ σ

σ + σ σ + σ

σ + σ

σ +

σ + σ σ + σ +

σ σ +

σ + σ σ + σ

− − + −

β ϕ =− ϕ

− − − − + − − ϕ − − − − + − +

− − (k 2)

(k 1)

(t )} ) σ + +

ϕ

(9)

Passing now to the estimation of the expression (8): For t∈[t_σ,t_σ+1] and t0∈ [tv, tv+1] with the conditions σ≠v-1,v+1 and v we obtain:

2 k ( 2 k 2)

N 1 M 1 ₀

0 k 0 t t

0

(2k 1) (2k 2)

2k (2k 1) 2k (2k 2) 2k

2k (2k 2)

(2k 1) (2k 1) 2k (2k 2) (2k 1)

2k (2k 1)

(t) (t )

1/( i)

t t

(t t )(t t )

{ (t )

(t t )(t t )

(t )

(t t )(t t )

(t

σ σ +

− −

σ= =

σ + σ +

σ

σ + σ σ + σ

σ σ +

σ +

σ + σ σ + σ +

σ σ +

σ

ϕ − ϕ π − − − − ϕ − − − − − ϕ − − − − +

∑ ∑ ∫

(2k 2) (2k 2) 2k (2k 2) (2k 1)

0 (2k 1) 0 (2k 2) 2k (2k 1) 2k (2k 2) 2k

0 2k 0 (2k 2)

(2k 1) (2k 1) 2k (2k 2) (2k 1)

0 2k 0 (2k 1)

(t )

t )(t t )

(t t )(t t )

(t )

(t t )(t t )

(t )

(t t )(t t )

(t

σ +

+ σ σ + σ +

ν + ν +

ν

ν + ν ν + ν

ν ν +

ν +

ν + ν ν + ν +

ν ν +

ν ϕ − − − − − ϕ − − − − + ϕ − − − − −

(2k 2) 2k (2k 2) (2k 1)

(2k 2) 0

ln 2MN

O( )

(2M) N

t )(t t )

1

(t )} dt

t t

µ µ

+ ν ν + ν +

ν +

=

− −

ϕ

−

Indeed, it is clear that:

( 2 k 2 ) 2 k 0 1

N 1M 1 _t

0 t

t t t 0 k 0 0

1 (t) (t ) ln 2MN

max | dt | O( )

i t t (2M) N

σ +

σ ν ν+

− −

µ µ

∈ σ= =

ϕ − φ ₌

π

∑∑∫

−

⌢

And:

( 2 k 2) 2 k

t N 1 M 1

0 k 0 t

( 2k 1) ( 2k 2)

2k ( 2k 1) 2k ( 2k 2) 2k

2k ( 2k 2)

(2k 1) ( 2k 1) 2k ( 2k 2) ( 2k 1)

2k (2k 1) (2k 2) 2k ( 2

1 i

(t t )(t t )

{ (t )

(t t )(t t )

(t )

(t t )(t t )

(t t )(t

σ +

σ

− − σ= =

σ + σ +

σ

σ + σ σ + σ

σ σ +

σ + σ + σ σ + σ +

σ σ +

σ + σ σ

π − − − ϕ − − − − − ϕ − − − − + −

∑ ∑ ∫

(2k 2) k 2) ( 2k 1)

0 ( 2k 1) 0 ( 2k 2) 2k ( 2k 1) 2k ( 2k 2) 2k

0 2k 0 (2k 2)

( 2k 1) ( 2k 1) 2k ( 2k 2) ( 2k 1)

0 2k 0 ( 2k 1) ( 2k 2) 2k (

(t )

t )

(t t )(t t )

(t ).

(t t )(t t )

(t )

(t t )(t t )

(t t )(t

σ + + σ +

ν + ν +

ν

ν + ν ν + ν

ν ν +

ν + ν + ν ν + ν +

ν ν +

ν + ν ν

ϕ − − − − ϕ − − − − + ϕ − − − − −

− 2k 2) (2k 1)

(2k 2) 0 ln 2MN O( ) (2M) N t ) 1

(t )} dt

t t

µ µ

+ ν +

ν +

=

−

ϕ

−

Further, for the three cases mentioned above σ = v-1,v+1 and v using the smoothness of the contour Γ and the function ϕ (t) in the Hölder space H (µ) we get:

1 1

s ₁

0

t t s

0

(t) (t )

| dt | A | s s | ds O(N )

t t

ν+

ν ν+ ν

µ− −µ

ϕ − ϕ _≤ ₋ ₌

−

∫

Example 1: Consider the singular integral:

0

1 (t)

I F(t ) dt

i Γt t

ϕ

= =

π

∫

−

where, Γ designates the circle centered at the origin point 0 with a unit radius and the density function ϕ is given by the following expression:

φ(t)=1/(t+2)

Table 1: Singular integral with φ(t)=1/(t+2)

N M ||I-Ip||1 ||I-Ip||2 ||I-I3||∞

10 3 4.484951E-04 2.2449576E-04 1.1798739E-04 10 4 2.5682151E-05 1.2854857E-05 6.8843365E-06 10 5 1.1920929E-06 7.300048E-07 5.9604645E-07

Example 2: We take the singular integral:

0

1 (t)

I F(t ) dt

i Γt t

ϕ

= =

π

∫

−

(4)

J. Math. & Stat., 8 (2): 292-295, 2012

295 2

t 1 (t)

t 5t 6

+ ϕ =

+ +

Table 2: Singular integral with 2

t 1 (t)

t 5t 6

+ ϕ =

+ +

N M ||I-Ip||1 ||I-Ip||2 ||I-I3||∞

10 3 3.6347099E-04 2.1240809E-04 1.7364323E-04 10 4 1.2820773E-04 7.9004079E-05 6.6116452E-05 10 5 6.1048195E-05 3.8276499E-05 3.1117350E-05

Numerical experiments: Using our approximation, we apply the algorithms to singular integrals and we present results concerning the accuracy of the calculations, in these numerical experiments each Table 1 and 2 represent the exact principal value of the singular integral and Ip corresponds to the approximate

calculation produced by our approximation at interpolation point’s values.

CONCLUSION

This approximation can be used to remove integrable singularity. It was tested for the numerical calculus of many singular integrals of Cauchy type, where it gave good results.

REFERENCES

Antidze, D.J., 1975. On the approximate solution of singular integral equations. Seminar Institute Applied Mathematics, Tbilissi,

Muskhelishvili, N.I., 2008. Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics. 2nd Edn., Dover Publications, New York, ISBN-10: 0486462420, pp: 464.

Nadir, M. and B. Lakehali, 2007. On the approximation of

singular integrals. FEN DERGĐSĐ (E-DERGĐ), 2: 236-240.

Nadir, M. and J. Antidze, 2004. On the numerical solution of singular integral equations using Sanikidze’s approximation. Comp. Meth. Sc Tech., 10: 83-89.

Nadir, M., 1985. Problèmes aux limites qui se reduisent aux equations intégrales de fredholm. Seminaire de

l’Institut de Mathématiques et Informatique, Annaba,

Nadir, M., 1998. Opérateurs intégraux et bases d’ondelettes. Far East J. Sci., 6: 977-995.